Algebraic Equations: An Introduction to the Theories of Lagrange and Galois 9780231877657

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ALGEBRAIC EQUATIONS

ALGEBRAIC EQUATIONS AN INTRODUCTION TO THE THEORIES OF LAGRANGE AND GALOIS

BY

EDGAR DEHN, PH.D. Instructor

of Mathematica

in Columbia

University

f l e w H>orft COLUMBIA UNIVERSITY PRESS 1930

COPYRIGHT COLUMBIA

1930

UNIVERSITY

PUBLISHED JANUARY

PRESS 1930

P R I N T E D IN T H E U N I T E D B T A T E S OR A M E R I C A THE UAPLE P R E S S COUPANY, YORK,

PA.

CONTENTS LIST

viii

OF T H E O R E M S

L I S T OF ADOPTED CONVENTIONS I.

II.

INTEGRAL

§1. §2. §3. §4. §5.

FUNCTION

Interpolation Division Reduction Primitive function Linear factors

E Q U A T I O N S AND

A L G E B R A OF

IV.

G R O U P AND

§9. §10. §11. §12. §13. §14. §15.

§16. §17. §18. §19. §20.

PERMUTATIONS

PERMUTATIONS

Notation Degree Combination Order Association Inverse Identity

11 13 15 17 18 19 21 23 24 26

SUBGROUP

Group Subgroup Conjugate subgroups Rule of transforms Normal subgroup

S Y M M E T R I C G R O U P AND I T S

§21. §22. §23. §24. §25. §26. §27. VI.

1 3 6 7 8

§6. Discovery of Lagrange §7. Solution of cubic §8. Connection with permutations

III.

V.

ix

27 29 32 34 36 FUNCTIONS

Generator Symmetric sum Computation of symmetric sum Another computation Resultant Resultant as determinant Discriminant

C O M P O S I T I O N OF S Y M M E T R I C

§28. Composition-series §29. Alternating function

38 40 41 44 46 47 50

GROUP

v

53 54

vi

CONTENTS §30. §31. §32. §33.

VII.

THEORY

§34. §35. §36. §37. §38. §39. §40. VIII.

IX.

X.

§41. §42. §43. §44. §45. §46. §47. §48.

EQUATIONS

Quadratic equation Cubic e q u a t i o n . . . . Cubic equation, Continued Cubic equation, Continued Cubic equation, Continued Biquadratic equation Biquadratic equation, Continued Biquadratic equation, Continued

MORE

56 59 61 64

OF L A G R A N G E

Resolvent equation Lagrange's Theorem Lagrange's Theorem, Continued Plan of Lagrange Lagrange's solvent Special case of solvents Limits of Lagrange's plan

GENERAL

ABOUT

GROUPS

§49. Isomorphic groups §50. Transitive group §51. Imprimitive group §52. Quotient-group §53. Subgroups of quotient-group §54. Maximum normal subgroup §55. Constancy of composition-factors §56. Abelian group §57. Theorem of Cauchy §58. Metacyclic group Note on abstract group

66 67 70 74 75 78 80 82 83 85 87 88 90 93 95 99 101 104 107 110 Ill 113 115 118 120 126

DOMAIN

§59. §60. §61. §62. §63. XI.

Alternating group Composition of S and A Subgroups of S and A Group on functions

Algebraic domain Algebraic domain, Continued Conjugate domains Conjugate domains, Continued Normal domain

T H E O R Y OF

§64. §65. §66. §67. §68.

GALOIS

Special equation Galoisian function Galoisian resolvent Galoisian group Properties of Galoisian group

129 131 133 135 136 139 140 143 145 146

CONTENTS §69. §70. §71. §72. §73. §74. §75. §76. XII.

SPECIAL

§77. §78. §79. §80. §81. §82. §83. §84. §85. §86. INDEX

Plan of Galois General equation Duality of plane Irreducible equation Applications Imprimitive equation Reduction of group Natural irrationality

vii 149 153 155 156 158 161 163 169

EQUATIONS

Abelian equation Cyclic equation Roots of unity Congruence Fermat's Theorem Cyclotomic equation Discriminant of cyclotomic equation Applications Metacyclic equation Quintic equation

171 172 175 176 179 182 184 185 190 195 201

LIST OF THEOREMS Theorem Page 1 4 2 4 3 8 4 5 5 6 6 « 7 7 8 7 9 (Eisenatein) 8 10 10 11 10 12 10 13 21 14 22 15 22 16 23 17 23 18 24 19 26 20 27 21 28 22 29 23 31 24 32 25 32 26 33 27 33 28 34 29 (Rule of transforms). 35 30 35 31 36 32 37 33 38 34 41 35 (Fund. th. of sym. functions) 43 36 47 37 50 38 51 39 53 40 55 41 56 42 56 43 57 44 57 45 58 46 61 47 61 48 62 49 64 50 66 51 (Lagrange) 67 52 72 53 73

Theorem Page 64 76 55 78 56 80 57 99 88 100 59 101 60 102 61 103 62 103 63 104 64 105 65 106 66 108 67 110 68 Ill 69 112 70 (Jordan-Holder) 113 71 118 72 118 73 116 74 116 75 116 76 116 77 116 78 117 79 118 80 118 81 118 82 119 83 (Cauchy) 119 84 121 85 123 86 124 87 128 88 126 89 (Cayley) 127 90 131 91 132 92 133 93 134 94 135 95 138 96 141 97 141 98 142 99 143 100 144 10 1 144 102 144 103 145 104 146 105 (Property 1 of Gal. group) 146 106 147

viii

Theorem Page 107 148 108 (Property 2 of Gal. group) 148 109 148 11 0 148 11 1 149 11 2 150 150 11 3 11 4 150 11 5 151 116 (Lagrange-Galois).. 151 11 7 151 11 8 153 11« 154 120 156 12 1 157 12 2 157 12 3 158 124 158 12 5 162 12 6 163 127 164 12 8 105 129 166 130 167 13 1 109 13 2 170 133 171 13 4 172 135 172 136 173 13 7 174 138 174 139 176 14 0 177 14 1 178 142 (Fermat) 179 14 3 180 14 4 180 14 5 182 14 6 182 14 7 183 14 8 183 14 9 184 150 (Galois) 190 15 1 191 15 2 191 15 3 192 154 193 15 5 193 15 6 195 15 7 195 15 8 196 159 200

LIST OF ADOPTED CONVENTIONS While it is neither possible nor desirable to stereotype algebraic notation, certain conventions are expedient if waived when inopportune.

S n

sum of functions product of functions argument of function roots of function function of x function of the Xi

X Xi,ai

f,g , . . . cn'-.

W = 4 permits

CI4,

D ^ 2 excludes ci s(x i2x22)

c22,

= c 2 2 + pcici +

(b) S(XI2X22X3)

=

^

CIC 3 ,

c4

4

qct.

P„CI"' . . . c n " - .

W = 5 permits CiC22, c2c3, cic4, c6 D ^ 2 excludes cic22 S ( X I 2 X 2 2 X 3 ) = c2c3 + PC1C4 + qc&. For computation of the numerical coefficients, which remains to be done and which is done best by the use of special equations, we refer to the examples of chapter eight. Also, there exist tables with the results of such computation. They are arranged according to the total degree of the symmetric sums, and we reproduce the table for the total degree four: c4 S(X!4) 3

s{Xi Xi) s(x , 2 X 2 2 ) S(XI2X2X3) S(XIX2X3X4)

-4 4 2 -4 1

C3c 1

4 -1 -2 1

c22

2 -2 1

C2Ci2

Ci4

-4 1

1

46

ALGEBRAIC

EQUATIONS

This table gives, for instance: ««>(x!«) = ci* - 4c'c, + 2c,* + 4clc, - 4c4 «'"(x. 4 ) = C!4 - 4cilc, + 2c,1. Since symmetric functions are composed of symmetric sums, the rules of computation may be applied to them directly, the highest weight and degree setting the mark. {25. RESULTANT*

Among the symmetric functions in the letters x,- which represent the roots of the general equation f(x)

= aox- + a , i - ' + . . . + a . = 0

are the resultant and the discriminant. If besides the function /(x) we have another such function g(x) = box" + 6,x"~l + . . . + bK, it will be convenient to denote the roots of /(x) by a, and the roots of g(x) by The two functions /(x) and g(x) have a root at in common if the product g{ai)g(at)

. . . giam)

vanishes with becoming zero. This product is evidently an integral function in the Since a permutation between the at only interchanges its factors, the product is a symmetric function of the cti and as such by proposition (35) rationally expressible in terms of o, a j

a„

i • • • »ao ao ao >

Since the product is of degree not more than n in any one an, it is of total degree not more than n in the Ci/aa. Multiplying it by a

and is an alternating function. Taking now a permutation t h a t alters , we resolve it into transpositions. As we successively apply these transpositions to we must strike one t h a t changes the sign of 3>, for the permutation does so. If it is (xix 2 ), so that (xi, Xt, ...

, x„) =

an, . . . , x n ),

we set Xi = xt;

this gives 4>(xi, . . . , ! „ ) =

. • • , X„) = 0,

since both members differ by their sign alone. It appears t h a t is divisible by an — x2. But then «Ê2 is divisible by (an — x 2 ) 2 and, as it evidently is symmetric, by every (Xi — xt) 2 . Hence $ in turn is divisible by every xt — x* and equals y / à . = n (x< -

Xt)

itself or multiplied by a symmetric function. A1 + At = Ai - At

Setting

= 2 Si = 2StVÂ,

we find that (40) the general form of alternating functions is Ai

= S! +

StVà

At

= Si -

StVÂ.

56

ALGEBRAIC

EQUATIONS

With the alternating functions exists the group of these functions which is called the alternating group and is denoted by A or, to guard against confusion, by| A): (41) The alternating group is a normal subgroup of index two in the symmetric group. It is the greatest subgroup of the symmetric group, containing one half of its permutations. The simplest function that belongs to it is the root V a of the discriminant. §30. ALTERNATING GROUP

Every transposition alters the sign of \ / a and the value of an alternating function, as we can verify on any example. Writing subscripta only, we have in five letters x¿, for instance: VA = (1 - 2)(1 - 3)(1 - 4)(1 (2 - 3) (2 - 4) (2 (3 - 4) (3 (4

-

5) 5) 5) 5).

By the transposition (24), say, the factors (1 — 2) and (2 — 5) only change place with (1 — 4) and (4 — 5), the factor (2 — 3) changes place with (3 — 4) and also sign. While this leaves VA unaltered, there is one factor, the factor (2 — 4) containing the numbers of the transposition, which changes its own sign and that of V a . Hence we conclude that any even number of transpositions leaves Va unaltered while any odd number of them alters its sign. If then some permutation once breaks up into an even number of transpositions, it always does so since it cannot alter and not alter the sign of V a . Such a permutation is called an even permutation; while another which is formed by an odd number of transpositions is called odd. (42) A circular permutation, or the cycle of a non-circular permutation, is odd or even according as its degree is even or odd; a non-circular permutation is odd or even according as it contains an odd or even number of odd cycles. For instance: (123) = (12)(13)

COMPOSITION OF SYMMETRIC

GROUP

57

is even while (1234) = (12)(13)(14) is odd, and (1234) (56) is even because it contains two odd cycles. Two similar permutations evidently are both odd or both even. It is clear that (43) the alternating group is composed of all even permutations in the symmetric group, all those that leave unaltered the sign of VA. We readily admit that such permutations form a group, for the product of even permutations is again an even permutation; also that this group is normal, for the transform of an even permutation is by proposition (30) again even. Every permutation on n letters x, can by proposition (33) be represented as the product of transpositions which are in the set (ziXi)^; every even permutation therefore as the product of two such transpositions, which is to say as the product of circular permutations (XtXiX*) = (XiZ,) (XiXjl ) of order three.

But we can easily verify that (XiXiXt) = {XiXiX^iXiXiXk),

whence the permutations of the alternating group are resolvable into cycles of order three and, if we so choose, into such as give the n — 2 permutations ( x ^ x * ) " = (X1X2X3), (X1X2X4), . . . , (XiX2X„). It follows that (44) the alternating group on n letters x< is generated by the n — 2 independent permutations ( j r a ^ ) " : A» = {(xixgjci)?)]; and a group including these permutations is the alternating if not the symmetric group.

58

ALGEBRAIC

EQUATIONS

For the lowest degrees the alternating groups are: A2 = 1 A3 = 1, (123), (132) A4 =

1

(12) (34)

(13)(24)

(14)(23)

(123) (132)

(243) (143)

(142) (234)

(134) (124)

If we denote the numbers 1, 2, 3, 4, . . . in any order whatever by til Î2) t'i, it, . . . we can assert that the two permutations ,,-( and

1 2 3 4 •••) \Î1Î2I3Î4 • • . /

1 2 3 4 «, = (\Î2ÎIÎ3Î4 •• •• •• /)

cannot both be odd or even, for their product trft is odd since the product W2"1 = (12) is odd and the permutations t2 and are similar by proposition (18). Hence either ti or h is in the alternating group, and (45) a normal subgroup of the alternating group contains with one circular permutation of order three every possible such permutation. For suppose it contains the permutation s = (123). If the permutation l = X2 + x 3 , = Xi + X3, ¿3 = X! + Xi. This verifies that a subgroup of index n in Sn is S"~l, the group for instance of 1

1 means 1

Bv proposition (46).

j ! k Pr = 1,

under A is

64

ALGEBRAIC

EQUATIONS

we have 2The exceptional case r = 4 when n = 4 gives j! ^ 6 and permits

1 < j = 3 < 4.

The same function k in

— W is homogeneous of But so is the determinant as

+ 0" " 1) =

JU ~ 1)

and their quotient can be only numerical. From the leading term of the determinant, as compared with the corresponding term of

—

it is seen to be 1, whence 1

1

1

1

=

Vaj,

^jj-l H - 1 i t ' - 1 . Denoting now by T the determinant of the numerator, we may set T. V a , Vi = A. Va, where A, is symmetric in the 4>i and hence the x t and as such rational in the c,. To complete our proof, it remains to investigate the numerator T . VL,

r„ ri

=

1

1

... ... ...

Ì//3

Ti

W

1

. Va,

ij

f, 2

i-i ^ ¡ r 1 tz'- 1 . . . tr1 It is a function symmetric in the conjugate values ii,

• • • , ii

other than \f>u for interchanging any two such values we alter the sign of both T and V a , but do not alter their product. To examine such a function, we take the equation -

)(*

-

.

.

.

=

-

+ A

+

A^-2 + . . . +Aj= 0

70

ALGEBRAIC

EQUATIONS

with coefficients rational in the ct and dividing out $ — $i obtain the equation =

+

where

+ Ai)^'- 1 + + Ai\pi + A,)*'"» + . . . = + B + + . . . = 0, Bl = fr + A,

B* = *i 2 + Arf-i + Ai Hence it appears that a symmetric function of the roots f < other than f i, rationally expressible in terms of the elementary symmetric functions of those roots, is expressible so in terms of and the A„ and therefore expressible so in terms of and the c.. Conversely, if i, cO,

then a permutation leaving unaltered leaves unaltered also , e{). It follows that the group G of a binomial resolvent2 either has a normal subgroup of prime index or a series of such: G J «-«-» N P-i

are interchanged in p different ways, as explained in §33, and by proposition (49) the permutations between these conjugate values form a group. This group is circular by proposition (25), and we may set r = 1, T, T2, . . . , T*-1 T = (¿oil . . . ^p_l). We now form the function (f,*)o = to + e* i + eVi + • • • + «"-Vp-2 + «"-Vp-i also belonging to N, where t = cos

2jt , . . 2ir h i sin —V

V

1

is a primitive root of unity and t" = 1. A permutation in the partition Nt\ interchanges the function as T indicates, therefore cyclically, converting

of the into

= + «¿2 + eVi + • • • + «"-Vp-i + «""Vo = *)o, while a permutation in the partition Nt 2 interchanges the \pi as r J indicates converting (t,^) 0 into =

and so on.

+ «*» + tS>t + . . .

It appears that the permutations of

G on the i , :

like those of r on the yf/i'. convert (e,^)

N

Nh

1

T

(*,*)>

o into: equal to:

M ) «

(«,*)«>

and t o : •Cf. | §79

+ t'-Vo +

and 84.

Nt, T*

M ) >

0

« " - ' ( « , 0

€»-'(€,,(r)o

t'-'h

78

ALGEBRAIC

EQUATIONS

if we multiply by t* = 1. As we have the function (t,^)op is unaltered by any permutation of G and such a permutation alone, so that by Lagrange's Theorem =

R(ci,,x)i, and such a function (t,